---
_id: '7944'
abstract:
- lang: eng
text: "This thesis considers two examples of reconfiguration problems: flipping
edges in edge-labelled triangulations of planar point sets and swapping labelled
tokens placed on vertices of a graph. In both cases the studied structures – all
the triangulations of a given point set or all token placements on a given graph
– can be thought of as vertices of the so-called reconfiguration graph, in which
two vertices are adjacent if the corresponding structures differ by a single elementary
operation – by a flip of a diagonal in a triangulation or by a swap of tokens
on adjacent vertices, respectively. We study the reconfiguration of one instance
of a structure into another via (shortest) paths in the reconfiguration graph.\r\n\r\nFor
triangulations of point sets in which each edge has a unique label and a flip
transfers the label from the removed edge to the new edge, we prove a polynomial-time
testable condition, called the Orbit Theorem, that characterizes when two triangulations
of the same point set lie in the same connected component of the reconfiguration
graph. The condition was first conjectured by Bose, Lubiw, Pathak and Verdonschot.
We additionally provide a polynomial time algorithm that computes a reconfiguring
flip sequence, if it exists. Our proof of the Orbit Theorem uses topological properties
of a certain high-dimensional cell complex that has the usual reconfiguration
graph as its 1-skeleton.\r\n\r\nIn the context of token swapping on a tree graph,
we make partial progress on the problem of finding shortest reconfiguration sequences.
We disprove the so-called Happy Leaf Conjecture and demonstrate the importance
of swapping tokens that are already placed at the correct vertices. We also prove
that a generalization of the problem to weighted coloured token swapping is NP-hard
on trees but solvable in polynomial time on paths and stars."
alternative_title:
- IST Austria Thesis
article_processing_charge: No
author:
- first_name: Zuzana
full_name: Masárová, Zuzana
id: 45CFE238-F248-11E8-B48F-1D18A9856A87
last_name: Masárová
orcid: 0000-0002-6660-1322
citation:
ama: Masárová Z. *Reconfiguration Problems*. IST Austria; 2020. doi:10.15479/AT:ISTA:7944
apa: Masárová, Z. (2020). *Reconfiguration problems*. IST Austria. https://doi.org/10.15479/AT:ISTA:7944
chicago: Masárová, Zuzana. *Reconfiguration Problems*. IST Austria, 2020. https://doi.org/10.15479/AT:ISTA:7944.
ieee: Z. Masárová, *Reconfiguration problems*. IST Austria, 2020.
ista: Masárová Z. 2020. Reconfiguration problems, IST Austria, 160p.
mla: Masárová, Zuzana. *Reconfiguration Problems*. IST Austria, 2020, doi:10.15479/AT:ISTA:7944.
short: Z. Masárová, Reconfiguration Problems, IST Austria, 2020.
date_created: 2020-06-08T00:49:46Z
date_published: 2020-06-09T00:00:00Z
date_updated: 2020-08-11T10:10:28Z
day: '09'
ddc:
- '516'
- '514'
department:
- _id: HeEd
- _id: UlWa
doi: 10.15479/AT:ISTA:7944
file:
- access_level: open_access
checksum: df688bc5a82b50baee0b99d25fc7b7f0
content_type: application/pdf
creator: zmasarov
date_created: 2020-06-08T00:34:00Z
date_updated: 2020-07-14T12:48:05Z
file_id: '7945'
file_name: THESIS_Zuzka_Masarova.pdf
file_size: 13661779
relation: main_file
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checksum: 45341a35b8f5529c74010b7af43ac188
content_type: application/zip
creator: zmasarov
date_created: 2020-06-08T00:35:30Z
date_updated: 2020-07-14T12:48:05Z
file_id: '7946'
file_name: THESIS_Zuzka_Masarova_SOURCE_FILES.zip
file_size: 32184006
relation: source_file
file_date_updated: 2020-07-14T12:48:05Z
has_accepted_license: '1'
keyword:
- reconfiguration
- reconfiguration graph
- triangulations
- flip
- constrained triangulations
- shellability
- piecewise-linear balls
- token swapping
- trees
- coloured weighted token swapping
language:
- iso: eng
license: https://creativecommons.org/licenses/by-sa/4.0/
month: '06'
oa: 1
oa_version: Published Version
page: '160'
publication_identifier:
eissn:
- 2663-337X
isbn:
- 978-3-99078-005-3
publication_status: published
publisher: IST Austria
related_material:
record:
- id: '7950'
relation: part_of_dissertation
status: public
- id: '5986'
relation: part_of_dissertation
status: public
status: public
supervisor:
- first_name: Uli
full_name: Wagner, Uli
id: 36690CA2-F248-11E8-B48F-1D18A9856A87
last_name: Wagner
orcid: 0000-0002-1494-0568
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
title: Reconfiguration problems
tmp:
image: /images/cc_by_sa.png
legal_code_url: https://creativecommons.org/licenses/by-sa/4.0/legalcode
name: Creative Commons Attribution-ShareAlike 4.0 International Public License (CC
BY-SA 4.0)
short: CC BY-SA (4.0)
type: dissertation
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2020'
...
---
_id: '8317'
abstract:
- lang: eng
text: When can a polyomino piece of paper be folded into a unit cube? Prior work
studied tree-like polyominoes, but polyominoes with holes remain an intriguing
open problem. We present sufficient conditions for a polyomino with one or several
holes to fold into a cube, and conditions under which cube folding is impossible.
In particular, we show that all but five special “basic” holes guarantee foldability.
acknowledgement: This research was performed in part at the 33rd Bellairs Winter Workshop
on Computational Geometry. We thank all other participants for a fruitful atmosphere.
H. Akitaya was supported by NSF CCF-1422311 & 1423615. Z. Masárová was partially
funded by Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31.
article_number: '101700'
article_processing_charge: No
article_type: original
author:
- first_name: Oswin
full_name: Aichholzer, Oswin
last_name: Aichholzer
- first_name: Hugo A.
full_name: Akitaya, Hugo A.
last_name: Akitaya
- first_name: Kenneth C.
full_name: Cheung, Kenneth C.
last_name: Cheung
- first_name: Erik D.
full_name: Demaine, Erik D.
last_name: Demaine
- first_name: Martin L.
full_name: Demaine, Martin L.
last_name: Demaine
- first_name: Sándor P.
full_name: Fekete, Sándor P.
last_name: Fekete
- first_name: Linda
full_name: Kleist, Linda
last_name: Kleist
- first_name: Irina
full_name: Kostitsyna, Irina
last_name: Kostitsyna
- first_name: Maarten
full_name: Löffler, Maarten
last_name: Löffler
- first_name: Zuzana
full_name: Masárová, Zuzana
id: 45CFE238-F248-11E8-B48F-1D18A9856A87
last_name: Masárová
orcid: 0000-0002-6660-1322
- first_name: Klara
full_name: Mundilova, Klara
last_name: Mundilova
- first_name: Christiane
full_name: Schmidt, Christiane
last_name: Schmidt
citation:
ama: 'Aichholzer O, Akitaya HA, Cheung KC, et al. Folding polyominoes with holes
into a cube. *Computational Geometry: Theory and Applications*. 93. doi:10.1016/j.comgeo.2020.101700'
apa: 'Aichholzer, O., Akitaya, H. A., Cheung, K. C., Demaine, E. D., Demaine, M.
L., Fekete, S. P., … Schmidt, C. (n.d.). Folding polyominoes with holes into a
cube. *Computational Geometry: Theory and Applications*, *93*. https://doi.org/10.1016/j.comgeo.2020.101700'
chicago: 'Aichholzer, Oswin, Hugo A. Akitaya, Kenneth C. Cheung, Erik D. Demaine,
Martin L. Demaine, Sándor P. Fekete, Linda Kleist, et al. “Folding Polyominoes
with Holes into a Cube.” *Computational Geometry: Theory and Applications*
93 (n.d.). https://doi.org/10.1016/j.comgeo.2020.101700.'
ieee: 'O. Aichholzer *et al.*, “Folding polyominoes with holes into a cube,”
*Computational Geometry: Theory and Applications*, vol. 93.'
ista: 'Aichholzer O, Akitaya HA, Cheung KC, Demaine ED, Demaine ML, Fekete SP, Kleist
L, Kostitsyna I, Löffler M, Masárová Z, Mundilova K, Schmidt C. Folding polyominoes
with holes into a cube. Computational Geometry: Theory and Applications. 93.'
mla: 'Aichholzer, Oswin, et al. “Folding Polyominoes with Holes into a Cube.” *Computational
Geometry: Theory and Applications*, vol. 93, 101700, Elsevier, doi:10.1016/j.comgeo.2020.101700.'
short: 'O. Aichholzer, H.A. Akitaya, K.C. Cheung, E.D. Demaine, M.L. Demaine, S.P.
Fekete, L. Kleist, I. Kostitsyna, M. Löffler, Z. Masárová, K. Mundilova, C. Schmidt,
Computational Geometry: Theory and Applications 93 (n.d.).'
date_created: 2020-08-30T22:01:09Z
date_published: 2020-08-31T00:00:00Z
date_updated: 2020-09-10T10:09:40Z
day: '31'
department:
- _id: HeEd
doi: 10.1016/j.comgeo.2020.101700
external_id:
arxiv:
- 1910.09917v3
intvolume: ' 93'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1910.09917v3
month: '08'
oa: 1
oa_version: Preprint
project:
- _id: 268116B8-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: Z00342
name: The Wittgenstein Prize
publication: 'Computational Geometry: Theory and Applications'
publication_identifier:
issn:
- '09257721'
publication_status: inpress
publisher: Elsevier
quality_controlled: '1'
related_material:
record:
- id: '6989'
relation: shorter_version
status: public
scopus_import: '1'
status: public
title: Folding polyominoes with holes into a cube
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 93
year: '2020'
...
---
_id: '7950'
abstract:
- lang: eng
text: "The input to the token swapping problem is a graph with vertices v1, v2,
. . . , vn, and n tokens with labels 1,2, . . . , n, one on each vertex. The
goal is to get token i to vertex vi for all i= 1, . . . , n using a minimum number
of swaps, where a swap exchanges the tokens on the endpoints of an edge.Token
swapping on a tree, also known as “sorting with a transposition tree,” is not
known to be in P nor NP-complete. We present some partial results:\r\n1. An
optimum swap sequence may need to perform a swap on a leaf vertex that has the
correct token (a “happy leaf”), disproving a conjecture of Vaughan.\r\n2. Any
algorithm that fixes happy leaves—as all known approximation algorithms for the
problem do—has approximation factor at least 4/3. Furthermore, the two best-known
2-approximation algorithms have approximation factor exactly 2.\r\n3. A generalized
problem—weighted coloured token swapping—is NP-complete on trees, but solvable
in polynomial time on paths and stars. In this version, tokens and vertices
\ have colours, and colours have weights. The goal is to get every
token to a vertex of the same colour, and the cost of a swap is the sum of the
weights of the two tokens involved."
article_processing_charge: No
author:
- first_name: Ahmad
full_name: Biniaz, Ahmad
last_name: Biniaz
- first_name: Kshitij
full_name: Jain, Kshitij
last_name: Jain
- first_name: Anna
full_name: Lubiw, Anna
last_name: Lubiw
- first_name: Zuzana
full_name: Masárová, Zuzana
id: 45CFE238-F248-11E8-B48F-1D18A9856A87
last_name: Masárová
orcid: 0000-0002-6660-1322
- first_name: Tillmann
full_name: Miltzow, Tillmann
last_name: Miltzow
- first_name: Debajyoti
full_name: Mondal, Debajyoti
last_name: Mondal
- first_name: Anurag Murty
full_name: Naredla, Anurag Murty
last_name: Naredla
- first_name: Josef
full_name: Tkadlec, Josef
id: 3F24CCC8-F248-11E8-B48F-1D18A9856A87
last_name: Tkadlec
- first_name: Alexi
full_name: Turcotte, Alexi
last_name: Turcotte
citation:
ama: Biniaz A, Jain K, Lubiw A, et al. Token swapping on trees. *arXiv:190306981*.
apa: Biniaz, A., Jain, K., Lubiw, A., Masárová, Z., Miltzow, T., Mondal, D., … Turcotte,
A. (n.d.). Token swapping on trees. *ArXiv:1903.06981*. ArXiv.
chicago: Biniaz, Ahmad, Kshitij Jain, Anna Lubiw, Zuzana Masárová, Tillmann Miltzow,
Debajyoti Mondal, Anurag Murty Naredla, Josef Tkadlec, and Alexi Turcotte. “Token
Swapping on Trees.” *ArXiv:1903.06981*. ArXiv, n.d.
ieee: A. Biniaz *et al.*, “Token swapping on trees,” *arXiv:1903.06981*.
ArXiv.
ista: Biniaz A, Jain K, Lubiw A, Masárová Z, Miltzow T, Mondal D, Naredla AM, Tkadlec
J, Turcotte A. Token swapping on trees. arXiv:1903.06981.
mla: Biniaz, Ahmad, et al. “Token Swapping on Trees.” *ArXiv:1903.06981*, ArXiv.
short: A. Biniaz, K. Jain, A. Lubiw, Z. Masárová, T. Miltzow, D. Mondal, A.M. Naredla,
J. Tkadlec, A. Turcotte, ArXiv:1903.06981 (n.d.).
date_created: 2020-06-08T12:25:25Z
date_published: 2019-03-16T00:00:00Z
date_updated: 2020-07-14T23:17:23Z
day: '16'
department:
- _id: HeEd
- _id: UlWa
- _id: KrCh
external_id:
arxiv:
- '1903.06981'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1903.06981
month: '03'
oa: 1
oa_version: Preprint
page: '41'
publication: arXiv:1903.06981
publication_status: submitted
publisher: ArXiv
related_material:
record:
- id: '7944'
relation: dissertation_contains
status: public
status: public
title: Token swapping on trees
type: preprint
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2019'
...
---
_id: '5986'
abstract:
- lang: eng
text: "Given a triangulation of a point set in the plane, a flip deletes an edge
e whose removal leaves a convex quadrilateral, and replaces e by the opposite
diagonal of the quadrilateral. It is well known that any triangulation of a point
set can be reconfigured to any other triangulation by some sequence of flips.
We explore this question in the setting where each edge of a triangulation has
a label, and a flip transfers the label of the removed edge to the new edge. It
is not true that every labelled triangulation of a point set can be reconfigured
to every other labelled triangulation via a sequence of flips, but we characterize
when this is possible. There is an obvious necessary condition: for each label
l, if edge e has label l in the first triangulation and edge f has label l in
the second triangulation, then there must be some sequence of flips that moves
label l from e to f, ignoring all other labels. Bose, Lubiw, Pathak and Verdonschot
formulated the Orbit Conjecture, which states that this necessary condition is
also sufficient, i.e. that all labels can be simultaneously mapped to their destination
if and only if each label individually can be mapped to its destination. We prove
this conjecture. Furthermore, we give a polynomial-time algorithm (with \U0001D442(\U0001D45B8)
being a crude bound on the run-time) to find a sequence of flips to reconfigure
one labelled triangulation to another, if such a sequence exists, and we prove
an upper bound of \U0001D442(\U0001D45B7) on the length of the flip sequence.
Our proof uses the topological result that the sets of pairwise non-crossing edges
on a planar point set form a simplicial complex that is homeomorphic to a high-dimensional
ball (this follows from a result of Orden and Santos; we give a different proof
based on a shelling argument). The dual cell complex of this simplicial ball,
called the flip complex, has the usual flip graph as its 1-skeleton. We use properties
of the 2-skeleton of the flip complex to prove the Orbit Conjecture."
article_processing_charge: Yes (via OA deal)
article_type: original
author:
- first_name: Anna
full_name: Lubiw, Anna
last_name: Lubiw
- first_name: Zuzana
full_name: Masárová, Zuzana
id: 45CFE238-F248-11E8-B48F-1D18A9856A87
last_name: Masárová
orcid: 0000-0002-6660-1322
- first_name: Uli
full_name: Wagner, Uli
id: 36690CA2-F248-11E8-B48F-1D18A9856A87
last_name: Wagner
orcid: 0000-0002-1494-0568
citation:
ama: Lubiw A, Masárová Z, Wagner U. A proof of the orbit conjecture for flipping
edge-labelled triangulations. *Discrete & Computational Geometry*. 2019;61(4):880-898.
doi:10.1007/s00454-018-0035-8
apa: Lubiw, A., Masárová, Z., & Wagner, U. (2019). A proof of the orbit conjecture
for flipping edge-labelled triangulations. *Discrete & Computational Geometry*,
*61*(4), 880–898. https://doi.org/10.1007/s00454-018-0035-8
chicago: 'Lubiw, Anna, Zuzana Masárová, and Uli Wagner. “A Proof of the Orbit Conjecture
for Flipping Edge-Labelled Triangulations.” *Discrete & Computational Geometry*
61, no. 4 (2019): 880–98. https://doi.org/10.1007/s00454-018-0035-8.'
ieee: A. Lubiw, Z. Masárová, and U. Wagner, “A proof of the orbit conjecture for
flipping edge-labelled triangulations,” *Discrete & Computational Geometry*,
vol. 61, no. 4, pp. 880–898, 2019.
ista: Lubiw A, Masárová Z, Wagner U. 2019. A proof of the orbit conjecture for flipping
edge-labelled triangulations. Discrete & Computational Geometry. 61(4), 880–898.
mla: Lubiw, Anna, et al. “A Proof of the Orbit Conjecture for Flipping Edge-Labelled
Triangulations.” *Discrete & Computational Geometry*, vol. 61, no. 4,
Springer Nature, 2019, pp. 880–98, doi:10.1007/s00454-018-0035-8.
short: A. Lubiw, Z. Masárová, U. Wagner, Discrete & Computational Geometry 61
(2019) 880–898.
date_created: 2019-02-14T11:54:08Z
date_published: 2019-06-01T00:00:00Z
date_updated: 2020-08-11T10:10:43Z
day: '01'
ddc:
- '000'
department:
- _id: UlWa
doi: 10.1007/s00454-018-0035-8
external_id:
arxiv:
- '1710.02741'
file:
- access_level: open_access
checksum: e1bff88f1d77001b53b78c485ce048d7
content_type: application/pdf
creator: dernst
date_created: 2019-02-14T11:57:22Z
date_updated: 2020-07-14T12:47:14Z
file_id: '5988'
file_name: 2018_DiscreteGeometry_Lubiw.pdf
file_size: 556276
relation: main_file
file_date_updated: 2020-07-14T12:47:14Z
has_accepted_license: '1'
intvolume: ' 61'
issue: '4'
language:
- iso: eng
month: '06'
oa: 1
oa_version: Published Version
page: 880-898
project:
- _id: B67AFEDC-15C9-11EA-A837-991A96BB2854
name: IST Austria Open Access Fund
publication: Discrete & Computational Geometry
publication_identifier:
issn:
- 0179-5376
- 1432-0444
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
related_material:
record:
- id: '7944'
relation: dissertation_contains
status: public
- id: '683'
relation: earlier_version
status: public
scopus_import: 1
status: public
title: A proof of the orbit conjecture for flipping edge-labelled triangulations
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 61
year: '2019'
...
---
_id: '6989'
abstract:
- lang: eng
text: 'When can a polyomino piece of paper be folded into a unit cube? Prior work
studied tree-like polyominoes, but polyominoes with holes remain an intriguing
open problem. We present sufficient conditions for a polyomino with hole(s) to
fold into a cube, and conditions under which cube folding is impossible. In particular,
we show that all but five special simple holes guarantee foldability. '
acknowledgement: This research was performed in part at the 33rd BellairsWinter Workshop on Computational Geometry. Wethank
all other participants for a fruitful atmosphere.
article_processing_charge: No
author:
- first_name: Oswin
full_name: Aichholzer, Oswin
last_name: Aichholzer
- first_name: Hugo A
full_name: Akitaya, Hugo A
last_name: Akitaya
- first_name: Kenneth C
full_name: Cheung, Kenneth C
last_name: Cheung
- first_name: Erik D
full_name: Demaine, Erik D
last_name: Demaine
- first_name: Martin L
full_name: Demaine, Martin L
last_name: Demaine
- first_name: Sandor P
full_name: Fekete, Sandor P
last_name: Fekete
- first_name: Linda
full_name: Kleist, Linda
last_name: Kleist
- first_name: Irina
full_name: Kostitsyna, Irina
last_name: Kostitsyna
- first_name: Maarten
full_name: Löffler, Maarten
last_name: Löffler
- first_name: Zuzana
full_name: Masárová, Zuzana
id: 45CFE238-F248-11E8-B48F-1D18A9856A87
last_name: Masárová
orcid: 0000-0002-6660-1322
- first_name: Klara
full_name: Mundilova, Klara
last_name: Mundilova
- first_name: Christiane
full_name: Schmidt, Christiane
last_name: Schmidt
citation:
ama: 'Aichholzer O, Akitaya HA, Cheung KC, et al. Folding polyominoes with holes
into a cube. In: *Proceedings of the 31st Canadian Conference on Computational
Geometry*. Canadian Conference on Computational Geometry; 2019:164-170.'
apa: 'Aichholzer, O., Akitaya, H. A., Cheung, K. C., Demaine, E. D., Demaine, M.
L., Fekete, S. P., … Schmidt, C. (2019). Folding polyominoes with holes into a
cube. In *Proceedings of the 31st Canadian Conference on Computational Geometry*
(pp. 164–170). Edmonton, Canada: Canadian Conference on Computational Geometry.'
chicago: Aichholzer, Oswin, Hugo A Akitaya, Kenneth C Cheung, Erik D Demaine, Martin
L Demaine, Sandor P Fekete, Linda Kleist, et al. “Folding Polyominoes with Holes
into a Cube.” In *Proceedings of the 31st Canadian Conference on Computational
Geometry*, 164–70. Canadian Conference on Computational Geometry, 2019.
ieee: O. Aichholzer *et al.*, “Folding polyominoes with holes into a cube,”
in *Proceedings of the 31st Canadian Conference on Computational Geometry*,
Edmonton, Canada, 2019, pp. 164–170.
ista: 'Aichholzer O, Akitaya HA, Cheung KC, Demaine ED, Demaine ML, Fekete SP, Kleist
L, Kostitsyna I, Löffler M, Masárová Z, Mundilova K, Schmidt C. 2019. Folding
polyominoes with holes into a cube. Proceedings of the 31st Canadian Conference
on Computational Geometry. CCCG: Canadian Conference in Computational Geometry
164–170.'
mla: Aichholzer, Oswin, et al. “Folding Polyominoes with Holes into a Cube.” *Proceedings
of the 31st Canadian Conference on Computational Geometry*, Canadian Conference
on Computational Geometry, 2019, pp. 164–70.
short: O. Aichholzer, H.A. Akitaya, K.C. Cheung, E.D. Demaine, M.L. Demaine, S.P.
Fekete, L. Kleist, I. Kostitsyna, M. Löffler, Z. Masárová, K. Mundilova, C. Schmidt,
in:, Proceedings of the 31st Canadian Conference on Computational Geometry, Canadian
Conference on Computational Geometry, 2019, pp. 164–170.
conference:
end_date: 2019-08-10
location: Edmonton, Canada
name: 'CCCG: Canadian Conference in Computational Geometry'
start_date: 2019-08-08
date_created: 2019-11-04T16:46:11Z
date_published: 2019-08-01T00:00:00Z
date_updated: 2020-09-10T10:09:39Z
day: '01'
department:
- _id: HeEd
external_id:
arxiv:
- 1910.09917v3
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://cccg.ca/proceedings/2019/proceedings.pdf
month: '08'
oa: 1
oa_version: Published Version
page: 164-170
publication: Proceedings of the 31st Canadian Conference on Computational Geometry
publication_status: published
publisher: Canadian Conference on Computational Geometry
quality_controlled: '1'
related_material:
record:
- id: '8317'
relation: extended_version
status: public
scopus_import: '1'
status: public
title: Folding polyominoes with holes into a cube
type: conference
user_id: D865714E-FA4E-11E9-B85B-F5C5E5697425
year: '2019'
...
---
_id: '683'
abstract:
- lang: eng
text: 'Given a triangulation of a point set in the plane, a flip deletes an edge
e whose removal leaves a convex quadrilateral, and replaces e by the opposite
diagonal of the quadrilateral. It is well known that any triangulation of a point
set can be reconfigured to any other triangulation by some sequence of flips.
We explore this question in the setting where each edge of a triangulation has
a label, and a flip transfers the label of the removed edge to the new edge. It
is not true that every labelled triangulation of a point set can be reconfigured
to every other labelled triangulation via a sequence of flips, but we characterize
when this is possible. There is an obvious necessary condition: for each label
l, if edge e has label l in the first triangulation and edge f has label l in
the second triangulation, then there must be some sequence of flips that moves
label l from e to f, ignoring all other labels. Bose, Lubiw, Pathak and Verdonschot
formulated the Orbit Conjecture, which states that this necessary condition is
also sufficient, i.e. that all labels can be simultaneously mapped to their destination
if and only if each label individually can be mapped to its destination. We prove
this conjecture. Furthermore, we give a polynomial-time algorithm to find a sequence
of flips to reconfigure one labelled triangulation to another, if such a sequence
exists, and we prove an upper bound of O(n7) on the length of the flip sequence.
Our proof uses the topological result that the sets of pairwise non-crossing edges
on a planar point set form a simplicial complex that is homeomorphic to a high-dimensional
ball (this follows from a result of Orden and Santos; we give a different proof
based on a shelling argument). The dual cell complex of this simplicial ball,
called the flip complex, has the usual flip graph as its 1-skeleton. We use properties
of the 2-skeleton of the flip complex to prove the Orbit Conjecture.'
alternative_title:
- LIPIcs
article_number: '49'
author:
- first_name: Anna
full_name: Lubiw, Anna
last_name: Lubiw
- first_name: Zuzana
full_name: Masárová, Zuzana
id: 45CFE238-F248-11E8-B48F-1D18A9856A87
last_name: Masárová
orcid: 0000-0002-6660-1322
- first_name: Uli
full_name: Wagner, Uli
id: 36690CA2-F248-11E8-B48F-1D18A9856A87
last_name: Wagner
orcid: 0000-0002-1494-0568
citation:
ama: 'Lubiw A, Masárová Z, Wagner U. A proof of the orbit conjecture for flipping
edge labelled triangulations. In: Vol 77. Schloss Dagstuhl - Leibniz-Zentrum für
Informatik; 2017. doi:10.4230/LIPIcs.SoCG.2017.49'
apa: 'Lubiw, A., Masárová, Z., & Wagner, U. (2017). A proof of the orbit conjecture
for flipping edge labelled triangulations (Vol. 77). Presented at the SoCG: Symposium
on Computational Geometry, Brisbane, Australia: Schloss Dagstuhl - Leibniz-Zentrum
für Informatik. https://doi.org/10.4230/LIPIcs.SoCG.2017.49'
chicago: Lubiw, Anna, Zuzana Masárová, and Uli Wagner. “A Proof of the Orbit Conjecture
for Flipping Edge Labelled Triangulations,” Vol. 77. Schloss Dagstuhl - Leibniz-Zentrum
für Informatik, 2017. https://doi.org/10.4230/LIPIcs.SoCG.2017.49.
ieee: 'A. Lubiw, Z. Masárová, and U. Wagner, “A proof of the orbit conjecture for
flipping edge labelled triangulations,” presented at the SoCG: Symposium on Computational
Geometry, Brisbane, Australia, 2017, vol. 77.'
ista: 'Lubiw A, Masárová Z, Wagner U. 2017. A proof of the orbit conjecture for
flipping edge labelled triangulations. SoCG: Symposium on Computational Geometry,
LIPIcs, vol. 77.'
mla: Lubiw, Anna, et al. *A Proof of the Orbit Conjecture for Flipping Edge Labelled
Triangulations*. Vol. 77, 49, Schloss Dagstuhl - Leibniz-Zentrum für Informatik,
2017, doi:10.4230/LIPIcs.SoCG.2017.49.
short: A. Lubiw, Z. Masárová, U. Wagner, in:, Schloss Dagstuhl - Leibniz-Zentrum
für Informatik, 2017.
conference:
end_date: 2017-07-07
location: Brisbane, Australia
name: 'SoCG: Symposium on Computational Geometry'
start_date: 2017-07-04
date_created: 2018-12-11T11:47:54Z
date_published: 2017-06-01T00:00:00Z
date_updated: 2020-08-11T10:10:43Z
day: '01'
ddc:
- '514'
- '516'
department:
- _id: UlWa
doi: 10.4230/LIPIcs.SoCG.2017.49
file:
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checksum: 24fdde981cc513352a78dcf9b0660ae9
content_type: application/pdf
creator: system
date_created: 2018-12-12T10:17:12Z
date_updated: 2020-07-14T12:47:41Z
file_id: '5265'
file_name: IST-2017-896-v1+1_LIPIcs-SoCG-2017-49.pdf
file_size: 710007
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file_date_updated: 2020-07-14T12:47:41Z
has_accepted_license: '1'
intvolume: ' 77'
language:
- iso: eng
month: '06'
oa: 1
oa_version: Published Version
publication_status: published
publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik
publist_id: '7033'
pubrep_id: '896'
quality_controlled: '1'
related_material:
record:
- id: '5986'
relation: later_version
status: public
scopus_import: 1
status: public
title: A proof of the orbit conjecture for flipping edge labelled triangulations
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: conference
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 77
year: '2017'
...